deterministic optimization

7/29/2019 Deterministic Optimization

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AN INTRODUCTIONTO DETERMINISTICOPTIMIZATION

Boni Sena, 2013

Various types of LP Linear Programming (LP)


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LECTURER

Instructor : Boni Sena

Contact : 0856-9-2423-611 (Only forclassrepresentative by phone or SMS)

Email : [email protected]

Web : binusmaya.binus.ac.id

Text book :

Ernest F. Haussler, Richard S. Paul, Richard J. Wood,Introductory Mathematical Analysis, Prentice Hall

P. Rama Murthy, Operations Research, New Age

International Frederick S. Hiller, Gerald J. Lieberman, Introduction to

operations research, Mc-Graw Hill

Hamdy A. Taha, Operations research : an introduction,

Pearson Prentice Hall Boni Sena, 2013
mailto:[email protected]:[email protected]


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YOUARE ...

Boni Sena, 2013

Each of you is responsible for your own decision


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CLASSISLIKE..

Boni Sena, 2013

We need some regulations to achieve our goal..


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TOACHIEVEOURGOALS.. Our class will be started at 11:10 WIB The last attendance allowed 20 minute after the class is

started

Students are responsible for all announcement,

assignment, quiz and exam Students must show courtesy and respect to your

friends and speakerduring the class

Dont working on other subject, sleeping, any informal

chatter or activity not related to the lectureAny usage ofcell-phone/blackberry is strictly prohibited

during class period

No late assignments accepted

Boni Sena, 2013


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POINTOF EVALUATION Exams (70%)

Exam 1: 35% (midterm) Exam 2: 35% (final)

Assignment (30%) Project: 15% Quizzes: 15% (Quiz will be held every two weeks)

No make up will be given under any circumstances

REGRADE POLICY

No regrade request will be accepted more than one week after a

paper has been returned Do not write on the original document, but turn in a separate

Grade Groveling form with what you want regraded and why

Ifno additional points given, then 20 points will be deducted

Boni Sena, 2013


7/27 Boni Sena, 2013

Problem ?

Governor of Jakarta want to make MRT. Can you help them to design

the track of MRT ?


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TRAFFIC JAM

Boni Sena, 2013

How to solve traffic jam ?


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Car Assembly LineHow to weld all the weld beads within minimum time ?


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BRAIN CANCER TREATMENT [1]

Boni Sena, 2013

Where to deploy radiotheraphy to maximize impactOn carcinogenic cells and minimize damage inOther cells ?


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Mathematics

Model/linear programming

Algorithm

Problem

Maximize the benefit/achieve the goal

Minimize the effort

The best decision

The best decision


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MODELOF DETERMINISTIC OPTIMIZATION [6]

Actual system

Simplifiedsystem

Boni Sena, 2013

Model

Mathematics

Stochastic

Statistic

LinearProgramming


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LINEAR PROGRAMMING Linear Programming A model which is used for solving optimization

problems under such assumptions as certainty, linearity, fixedtechnology and constantprofit per unit [8]

Boni Sena, 2013

Objective function

Constraints

DecisionVariable

Component of

LP [2]


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LP MODEL FORMULATION

Decision variables mathematical symbols representing levels of activity of an

operation

Objective function

a linear relationship reflecting the objective of an operation most frequent objective of business firms is to maximize

profit

most frequent objective of individual operational units(such as a production or packaging department) is tominimize cost

Constraint a linear relationship representing a restriction on decision

making


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Boni Sena, 2013

TYPESOF LP [2]


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TYPESOF LP (CONT.)

Boni Sena, 2013


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TYPESOF LP (CONT.)

Boni Sena, 2013


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MATHEMATICAL MODELOF LP [2]

Boni Sena, 2013

Max/min z = c1x1 + c2x2 + ... + cnxn

subject to constraints :

a11x1 + a12x2 + ... + a1nxn(, =, ) b1a21x1 + a22x2 + ... + a2nxn(, =, ) b2

:

am1x1 + am2x2 + ... + amnxn(, =, ) bm

xj = decision variablesbi = constraint levels

cj = objective function coefficients

aij = constraint coefficients


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EXAMPLEOF LP

Z= 4x1 + 7x2

Subject to

9 x1 + 4x2 100

7x1 + x2 60

x1 ,x2 0

Boni Sena, 2013

Objective function

Constraints


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STEP TO FORMULATEMODELOF LINEAR PROGRAMMING

1. Identify the decision variables

2. Quantify the decision consequences to bemaximized or minimized through ObjectiveFunction

3. What limits decisions? Formulate the constraints

Main constraints

Variable-type constraints (sign restriction)

Boni Sena, 2013


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MAXIMIZATION PROBLEM

Boni Sena, 2013

Labor Clay RevenuePRODUCT (hr/unit) (lb/unit) ($/unit)

Bowl 1 4 40Mug 2 3 50

There are 40 hours of labor and 120 pounds of clayavailable each day

Decision variablesx1 = number of bowls to produce

x2 = number of mugs to produce

RESOURCE REQUIREMENTS


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Boni Sena, 2013

LP Formulation: Example of maximization

Maximize Z= 40 x1 + 50 x2

Subject to

x1 + 2x2 40 hr (labor constraint)

4x1 + 3x2 120 lb (clay constraint)

x1 , x20


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MINIMIZATION PROBLEM

Boni Sena, 2013

CHEMICAL CONTRIBUTION

Brand Nitrogen (lb/bag) Phosphate (lb/bag)

Gro-plus 2 4

Crop-fast 4 3

Minimize the production cost to buy nitrogen and phosphate every day.The cost of nitrogen $ 6/day and the cost of phosphate $ 3/day.

There are 16 lb of Nitrogen and 24 lb of phosphate available each day

Decision variables

x1 = number of nitrogen

x2

= number of phosphate


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LP FORMULATION : EXAMPLEOFMINIMIZATION

Boni Sena, 2013

Minimize Z= $6x1 + $3x2

subject to

2x1 + 4x2 16 lb of nitrogen

4x1 + 3x2 24 lb of phosphatex1, x2 0


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REFERENCES

1.

Andrs Ramos, 2012. Operation Research and OptimizationTechniques. Lecture Notes. Universidad Pontificia Comillas

2. Beni Asllani. 2006. Linear Programming. Lecture Notes.University of Tennessee. John Wiley & Sons, Inc.

3. Ernest F. Haussler, Richard S. Paul, Richard J. Wood,

Introductory Mathematical Analysis, Prentice Hall4. Frederick S. Hiller, Gerald J. Lieberman, Introduction to

operations research, Mc-Graw Hill

5. Hamdy A. Taha. Operations research : an introduction,Pearson Prentice Hall

Boni Sena, 2013


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REFERENCES6. Jong Jek Siang. 2011. Riset Operasi Dalam Pendekatan

Algoritmis. Yogyakarta, Andi

7. K. Gita Ayu. 2011. An Introduction to deterministicoptimization. Lecture Notes. Bina Nusantara University.

(Available at binusmaya.binus.ac.id)8. P. Rama Murthy, Operations Research, New Age

International

9. Rosihan Asmara. Operation Research : LinearProgramming. Lecture Notes. Brawijaya University

B i S 2013

deterministic optimization

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